Half-logistic distribution

In probability theory and statistics, the half-logistic distribution is a continuous probability distribution&mdash;the distribution of the absolute value of a random variable following the logistic distribution. That is, for


 * $$X = |Y| \!$$

where Y is a logistic random variable, X is a half-logistic random variable.

Cumulative distribution function
The cumulative distribution function (cdf) of the half-logistic distribution is intimately related to the cdf of the logistic distribution. Formally, if F(k) is the cdf for the logistic distribution, then G(k) = 2F(k) &minus; 1 is the cdf of a half-logistic distribution. Specifically,


 * $$G(k) = \frac{1-e^{-k}}{1+e^{-k}} \text{ for } k\geq 0. \!$$

Probability density function
Similarly, the probability density function (pdf) of the half-logistic distribution is g(k) = 2f(k) if f(k) is the pdf of the logistic distribution. Explicitly,


 * $$g(k) = \frac{2 e^{-k}}{(1+e^{-k})^2} \text{ for } k\geq 0. \!$$